Given a Lawvere theory $T,$ is it possible to describe a Lawvere theory $\textrm{End}(T)$ such that $\textrm{End}(T)$-algebras describe "endomorphisms of $T$-algebras"?
In other words, what can be said about the monoid $\textrm{End}(A)$ if $A$ is a $T$-algebra?
Yes, for any Lawvere theory $\mathbb{T}$, there is a Lawvere theory classifying the endomorphisms of $\mathbb{T}$-algebras.
This can be constructed more generally as a tensor product of Lawvere theories: for any two Lawvere theories $\mathbb{T}_1$ and $\mathbb{T}_2$, there is a Lawvere theory $\mathbb{T}_1 \otimes \mathbb{T}_2$ such that $$ \mathbf{Alg}(\mathbb{T}_1 \otimes \mathbb{T}_2, \mathcal{E}) \cong \mathbf{Alg}(\mathbb{T}_1, \mathbf{Alg}(\mathbb{T}_2, \mathcal{E})) , $$ where $\mathbf{Alg}(\mathbb{T},\mathcal{E})$ is the category of internal $\mathbb{T}$-algebras in the category $\mathcal{E}$.
Write $\mathbb{T}_{\mathsf{End}}$ for the Lawvere theory with a single sort and a single unary operation. Internal $\mathbb{T}_{\mathsf{End}}$-algebras in a category $\mathcal{E}$ are exactly endomorphisms in $\mathcal{E}$.
Then $(\mathbb{T}_{\mathsf{End}} \otimes \mathbb{T})$-algebras are endomorphisms in $\mathbf{Alg}(\mathbb{T},\mathbf{Set})$:
$$ \mathbf{Alg}(\mathbb{T}_{\mathsf{End}} \otimes \mathbb{T}, \mathbf{Set}) \cong \mathbf{Alg}(\mathbb{T}_{\mathsf{End}}, \mathbf{Alg}(\mathbb{T}, \mathbf{Set})) \cong \mathsf{End}(\mathbf{Alg}(\mathbb{T},\mathbf{Set})). $$
A reference is Algebra valued functors in general and tensor products in particular by P. Freyd.
The answer is yes. For every Lawvere theory $T$, there is a Lawvere theory whose algebras are endomorphisms of $T$-algebras. I'll show this by passing through the category of PROPs.
Note first that every (colored) Lawvere theory is in particular a (colored) PROP. This is easy to see from the definitions of a Lawvere theory and of a PROP, and is also proven as Proposition 4.2 in the paper Lawvere Categories as Composed PROPs by Bonchi et al. The idea is that if you're given a Lawvere theory then you map to a PROP where the required strict symmetric monoidal product of the PROP is the given cartesian product in the Lawvere theory.
Now that we know Lawvere theories are PROPs, we can write down the endomorphism PROP, as is done in 2.19 of On homotopy invariance for algebras over colored PROPs by Mark Johnson and Donald Yau, there denoted $E_X$. This PROP is equivalent to a Lawvere theory, since we again have the cartesian product, and the components of $E_X$ are exponentials (i.e., homs), so are still determined by a generic object (using the terminology from Theorem A in the linked nLab article on Lawvere theories) which namely say where the generic object goes in each component.
